1. x^p=e^x

Hi, how do I find the intersection point for this given that p is between and not including 2 and 3 ? Thanks !

2. Re: x^p=e^x

$x^p = e^x$

$e^{\ln(x) p} = e^x$

$\ln(x) p = x$

this has to either be solved numerically or the solution can be put in terms of the Lambert W function.

$x = -p W\left(-\dfrac 1 p\right)$

3. Re: x^p=e^x

Originally Posted by romsek
$x^p = e^x$

$e^{\ln(x) p} = e^x$

$\ln(x) p = x$

this has to either be solved numerically or the solution can be put in terms of the Lambert W function.

$x = -p W\left(-\dfrac 1 p\right)$
Hey, thanks! No.. sorry, I did get to that part, unsure how to solve now to ensure that p is between 2 and 3, so i'd appreciate if you could give me the steps so i can understand further from what yu did?
thank you so much ! you will be responsible for passing my exam ! I just dont understand the last line, although igto to the second last line ? so please explain ? also, how do you proceed to find p and x when x>0 ?

4. Re: x^p=e^x

Originally Posted by DiscreteMathHelp
Hey, thanks! No.. sorry, I did get to that part, unsure how to solve now to ensure that p is between 2 and 3, so i'd appreciate if you could give me the steps so i can understand further from what yu did?
thank you so much ! you will be responsible for passing my exam ! I just dont understand the last line, although igto to the second last line ? so please explain ? also, how do you proceed to find p and x when x>0 ?
Looking at the plot of this as you vary $p$ is instructive.

First off there are no real solutions for $x>0$ if $p < e$

If $p=e$ there is one real solution, $x=e$

If $p>e$ there are two real solutions

What class is this for? I'd be very surprised if you are expected to know what the Lambert W function is.

Most likely you are expected to solve this numerically, or graphically, or something.